The fact that a commutative ring has a natural topological space associated with it is a really interesting coincidence.

Moreover, in the answers, Pete L. Clark gives a list of other "really interesting coincidences" of algebraic objects having naturally associated topological spaces.

Is there a deeper explanation of the occurrence of these "really interesting coincidences"? It seems to suggest that the standard definition of "topological space" (collection of subsets, unions, intersections, blah blah), which somehow always seemed kind of a weird and artificial definition to me, has some kind of deeper significance or explanation, since it pops up everywhere...

Edit 1: I should clarify a bit. Let me be more explicit: Is there a unified explanation (mathematical ... or perhaps not) for why various algebraic (where "algebraic" is loosely defined) objects should have naturally associated topological spaces? Pete in the comments notes that he does not like the use of the word "coincidence" here --- but if these things are not coincidences, then what's the explanation?

Of course I do understand the intuitive idea behind the definition of "topological space", and how it abstracts for example the notions of "neighborhood" and "near" and "far". It is not surprising that the formalism of topological spaces is useful and ubiquitous in situations involving things like R^n, subsets of R^n, manifolds, metric spaces, simplicial complexes, CW complexes, etc.

However, when you start with algebraic objects and then get topological spaces out of them --- I find that surprising somehow because a priori there is not necessarily anything "geometric" or "topological" or "shape-y" or "neighborhood-y" going on.

Edit 2: Somebody has voted to close, saying this is "not a real question". I apologize for my imprecision and vagueness, but I still think this is a real question, for which real (mathematical) answers can conceivably exist.

For example, I'm hoping that maybe there is a theorem along the lines of something like:

Given an algebraic object A satisfying blah, define Spec(A) to be the set of blah-blahs of A such that blah-blah-blah. There is a natural topology on Spec(A), defined by [something]. When A is a commutative ring, this agrees with the Zariski topology on the prime spectrum. When A is a commutative C^* algebra, this agrees with the [is there a name?] topology on the Gelfand spectrum. When A is a Boolean algebra... When A is a commutative Banach ring... etc.

Of course, such a theorem, if such a theorem exists at all, would also need a definition of 'algebraic object'.

The article is Esquisse d'un Programme (Sketch of a Program), which is available on the Grothendieck Circle website: grothendieckcircle.org The relevant part is around page 22.
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Steven GubkinFeb 8 '10 at 13:32

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For what it's worth, I don't regard a functor from a category of "algebraic objects" (broadly construed) to the category Top of topological spaces as a coincidence. Is it exciting that there are highly nontrivial -- sometimes fully faithful -- functors from algebraic categories to Top? Definitely. Were the first such examples of this (by Stone) surprising to the mathematical community? Presumably (I wasn't there). But I don't like the term "coincidence" here, and I certainly did not use it myself.
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Pete L. ClarkFeb 8 '10 at 13:46

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I think the fact that there are so many topological structures on various things has less to do with the things and more to do with our desire to put a topological structure on anything we study. This is the first step of trying to think of something in geometric terms rather than purely algebraically. (As various answers below explain, while the axioms are confusing at first, they try to encode quite intuitive notions of "near" and "far". The reason they are confusing is, in my opinion, that people worked hard to make these notions as general as possible, and apply them to very weird space)
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Ilya GrigorievFeb 8 '10 at 18:23

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Ilya has a point; I think mathematicians like to convince themselves that what they are studying is universal but of course the priorities of human mathematicians don't remain uninfluenced by the history of their subject.
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Qiaochu YuanFeb 8 '10 at 19:53

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I think you are interpreting the phrase "put a topological structure on" too strictly. What I interpret Ilya to mean is something like "study the classifying space of a group" instead of "talking about topological groups."
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Qiaochu YuanFeb 9 '10 at 6:43

7 Answers
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I will take the question at face value, but not in the sense of justifying the definition.

A topological space is a convenient way of encoding, or perhaps better, organising, certain types of information. (Vague but true! I will give some instances. the data is sometimes `spatial' but more often than not, is not.)

Perhaps we should not think of spaces as 'god given' merely 'convenient', and there are variants that are more appropriate in various contexts.

A related question, coming from an old Shape Theorist (myself) is : when someone starts a theorem with 'Given a space $X$...', how is the space 'given'? As an algebraic topologist I sometimes need to use CW-complexes, but face the inconvenience that if I could give the CW structure precisely I could probably write down an algebraic model for its homotopy type precisely, and vice versa, so a good model is exactly the same as the one I started with. I hoped for more insight into what the space 'was' from my modelling. Giving the space is the end of the process, not the beginning. Strange. A space is a pseudo-visual way of thinking about 'data', which encodes important features, or at leastsome features that we can analyse, partially.

If someone gives me a compact subspace of $\mathbb{R}^n$, perhaps using some equations and inequalities, can I work out algebraic invariants of its homotopy type, rather than just its weak homotopy type? The answer will usually be no. Yet important properties of $C^*$ algebras on such a space, can sometimes be related to algebraic topological invariants of the homotopy type.

Spaces can arise as ways of encoding actual data as in topological data analysis, where there is a 'cloud' of data points and the practitioner is supposed to say something about the underlying space from which the data comes. There are finitely many data points, but no open sets given, they are for the data analyst to 'divine'.

Not all spatial data is conveniently modelled by spaces as such and directed spaces of various types have been proposed as models for changing data. Models for space-time are like this, but also models for concurrent systems.

Looking at finite topological spaces is again useful for encoding finite data (and I have rarely seen infinite amounts of data). For instance, relations between finite sets of data can be and are modelled in this way. Finite spaces give all homotopy types realisable by finite simplicial complexes. Finite spaces can be given precisely (provided they are not too big!) How do invariants of finite spaces appear in their structure? (Note the problem of infinite intersections does not arise here!!!)

At the other extreme, do we need points? Are locales not cleaner beasties and they can arise in lots of algebraic situations, again encoding algebraic information. Is a locale a space?

I repeat topological spaces are convenient, and in the examples you cite from algebraic geometry they happen to fit for good algebraic reasons. In other contexts they don't. Any Grothendieck topos looks like sheaves on a space, but the space involved will not usually be at all `nice' in the algebraic topological sense, so we use the topos and pretend it is a space, more or less.

Just as a data point: I was scared of topology until I learned about locales. At that point the light went on, and I said, "Oh, topologies are representations of Heyting algebras! Intuitionistic logic isn't scary, so I shouldn't be scared of topology, either!"
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Neel KrishnaswamiFeb 9 '10 at 9:04

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People may know of the book 'Topology via logic' by Steve Vickers. That is very relevant to this question and this comment.
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Tim PorterFeb 9 '10 at 11:39

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That's just the one I read! It's a charming little book.
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Neel KrishnaswamiFeb 9 '10 at 13:27

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Tim, you seem to take an anthropocentric view at first, but in the end you suggest that there might be external reasons too. This is sort of a chicken & egg question, but I would appreciate a clarification of your point of view. Do you think that topology is a nice way that we came up with to organize things in our environment, or that natural topological organization is something we grew to like enough that we feel a need to emphasize it whenever possible?
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François G. Dorais♦Feb 10 '10 at 5:46

My thoughts are, sort of, pragmatic, i.e. what works! When looking at something, we tend to use `models'. I think there is something 'out there' to model, but the models are not the same as the 'reality' we seek to understand. A lot of mathematics involves 'relations' in the non-technical as well as the technical sense. You can model relations in various ways including 'spatial' ones. Look at Chu spaces (Vaughan Pratt), or the old theorem of Dowker on simplicial complexes associated to relations. Those ideas help us organise things in a useful way, and are inherent in the idea of relation.
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Tim PorterFeb 10 '10 at 7:55

I think a reasonable partial explanation comes from universal algebra. The lattice Con(A) of congruences of an algebra is always a complete algebraic lattice. Therefore, it is meet continuous in the sense that $\bigvee_i a \wedge b_i = a \wedge \bigvee_i b_i$ whenever the $b_i$ form a directed family of congruences. When Con(A) happens to be finitely distributive, then one can drop the 'directed' requirement. In this case, Con(A) becomes a frame and it can thus be viewed as an abstract topological space (i.e. a locale). In fact, since Con(A) is algebraic the corresponding locale is always spatial and it always corresponds to a concrete spectral space.

In the case of a commutative ring A, the lattice Con(A) is isomorphic with the lattice Id(A) of ideals of A. The lattice Id(A) is not always distributive. (Though it is when A is a Prüfer domain and hence when A is a Dedekind domain, for example.) To remedy this, one looks at the radical ideals of A, which are always better behaved, to define the Zariski spectrum. In my humble opinion, the existence of radicals makes commutative rings very special among algebras.

Topological spaces could be crudely --- and I mean crudely --- divided into two kinds:

The geometric. These are the spaces that come up all over geometry and algebraic topology — manifolds, CW complexes, configuration spaces, CW complexes, etc. These are almost invariably Hausdorff, though there are plenty of compact Hausdorff spaces that are often thought of as "pathological", such as the Cantor set. I don't much like terms such as "nice space" and "pathological", because although they might be intended harmlessly, they sound to me a bit dismissive towards the second kind of space.

The spectra. I'm using this term loosely (and not in the sense of homotopy theory), but I mean things like the spectrum of a ring, the spectrum of a Boolean algebra (= a compact Hausdorff totally disconnected space), the maximal spectrum of a C*-algebra, and then things such as Julia sets, dynamical attractors, and solutions to iterated function systems, all of which have a spectrummy feel to me.

The spectra seem a bit unloved. No one denies the importance of, say, Spec of a commutative ring, but still, I reckon that most mathematicians subconsciously regard geometric spaces as the primary kind, and sometimes the spectra simply get swept away as "pathological".

+1+&epsilon; (The little extra is for the delightful term spectrummy.)
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François G. Dorais♦Feb 9 '10 at 0:30

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There is at least one additional kind: functional analysis topologies, especially all the various weak convergence topologies.
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Ilya GrigorievFeb 9 '10 at 3:42

Ah, good point. For a start, Banach spaces and Hilbert spaces are all around.
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Tom LeinsterFeb 9 '10 at 4:11

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Tom, this is probably hopelessly muddle-headed of me, but it struck me that some of your geometric examples have a "built up from simpler pieces feel", which could be "direct-limit-ish", and some of your spectrummy example have an "inverse-limit-ish" feel. Perhaps the latter, which tend to arise via contravariant functors, are indicating that it's the algebraic objects which are built out of simpler pieces; with colimits being taken to limits?
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Yemon ChoiFeb 9 '10 at 4:55

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There's also the topology on the integers in Fürstenberg's proof of the infinitude of primes.
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AnonymousFeb 9 '10 at 7:15

(I have deleted my original answer to this question because the question was changed in such a way that it made my answer irrelevant. I still think that my basic point was valid, and so am posting a new answer to make that point again. As my original post gained quite a few votes, I judge it not ethical to completely reword my answer but keep those votes. I am also making this answer "community wiki" not because I think anyone else should edit it but to remove it from the reputation/vote game.)

I think that the basic answer to this question is that there are connections between algebraic and topological things because we look for them. And we look for them because we have, in the past, found them useful. Something I continually (and I mean "continually", just ask one) tell my students is that mathematicians are fundamentally lazy. If we have a good theorem, we don't just use it for what it was first proved for, we look for other ways to use it, ways to extend it, ways to push it further than it was ever intended to be pushed. So if, as a topologist, I see the algebraists doing wonderful things with classifying and studying rings, then I'll do my best to make a ring out of my topological space so that I can steal (sorry, "use") their ideas and save myself a lot of bother. Thus: cohomology theory and the whole area of homotopy theory.

That the reverse is true is no surprise. Again, mathematicians are lazy so if we see a bridge with lots of useful stuff going in one direction, we ignore the "one way" signs and go the other way.

You could then ask "Why do the bridges exist at all?". Well, they don't always exist. Sometimes we can construct them and sometimes not. It feels a bit like you are looking at a bridge and say "Wow! Who would ever have thought of putting a bridge there?!" but ignoring all the stumps and collapsed half-bridges that litter the riverbank. Of course, one can ask about a specific bridge and ask why that one didn't collapse, but the question feels much more general than that.

So, in conclusion, that bridges exist is, I feel, more down to the downright mulishness of mathematicians determined to build a bridge wherever they can, regardless of how many collapses and "Pont d'Avignon"s they create in the process.

The above, clearly, works for any two areas of mathematics. Thinking particular of topological spaces, then I think that the questioner is missing the point of "near" and "far" a little when he says:

However, when you start with algebraic objects and then get topological spaces out of them --- I find that surprising somehow because a priori there is not necessarily anything "geometric" or "topological" or "shape-y" or "neighborhood-y" going on.

(I should point out that the "near" and "far" bit added in the question is in response to my original answer.)

Consider this scenario:

I have something I don't know anything about.

Can I find something out about something like it but simpler?

Yes! Great! But how do I measure which things are better approximations of my unknown thing than others?

Isn't that just what is going on in studying these algebraic objects? The language is fundamentally topological so there's no surprise at all that topological spaces result.

So as soon as an area of mathematics becomes interesting in that there are things that you can't figure out easily and simply then the question of finding enough approximations comes in and thus topology.

In conclusion of this second part, "interesting = topological" so making the (bizarre) assumption that algebra is interesting, it must thus be topological.

Perhaps one other reason we like to stick topologies on things is to make our problem/framework "more finitary" (the liking for compactness-type properties on various spectra-of-algebras, for instance).
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Yemon ChoiFeb 9 '10 at 11:19

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+1 for the bridges metaphor; I get that feeling a lot.
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Qiaochu YuanFeb 9 '10 at 16:09

I agree with everything except for the "interesting = topological". I think it's more like "geometric implies topological". I'm a geometric person myself, but I think there are also perfectly fine - and interesting - non-geometric ways to look at problems; elementary Galois theory comes to mind (I don't just mean the Fundamental Theorem, but the whole philosophy of factoring polynomials by thinking of field extensions).
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Ilya GrigorievFeb 9 '10 at 21:15

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I might admit to going just a little over the top at the end ...
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Loop SpaceFeb 9 '10 at 21:31

Yeah, but functors of points pull a lot of the geometry back out of geometry. =p.
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Harry GindiFeb 10 '10 at 2:01

From the perspective of locale theory, a topological space is nothing more than a model of the theory of finitary conjunctions and infinitary disjunctions (corresponding, perhaps, to the intuition that semidecidable propositions are closed under precisely these operations, and the idea that "open" is somehow just a geometric casting of "semidecidable"); that is, a topological space is little more than a lattice with finite meets and infinitary joins, the former distributing over the latter (i.e., a frame). It is perhaps not all that surprising that many algebraic structures should give rise to (complete) lattices satisfying this distributive property, is it? Well, that's a subjective judgement. Perhaps it isn't the best answer. But it's what I'm going with for now. [That is, it's not so surprising that many algebraic objects should give rise to frames; the more surprising thing is the realization that frames could be understood (contravariantly) in geometric terms in the first place.]

We can identify a Hausdorff space with its poset of open sets because every convergent (thanks JDH) ultrafilter converges uniquely to a point, so in fact, all of the "set data" of the space is contained in the poset of opens itself. We can define a lattice structure on this poset that takes the place of our algebra defined by intersections and unions. This is why the category of Hausdorff spaces is actually a category monadic over sets. Hausdorff spaces are actually totally described by their algebras, which seems pretty cool to me.

Some speculation on the question:

We have a natural poset structure on subobjects of algebraic objects, which at least gets us partway to having a topology.

Further, we can classify maps out of the space by looking at kernels of maps. There is also a canonical action for normal subgroups and ideals (product of normal subgroups and sum of ideals). These have the nice property that they are closed under this operation. They are also closed under intersections. This gives us a complete modular lattice on kernels. The interesting case about rings is that we have a third operation, the product of ideals. The interesting thing about the product of ideals is that it is only defined for finite products. Then we have a somewhat natural structure to start working in (at least for rings).

I believe what Qiaochu was talking about with the Galois connections is that for any algebraic structure, you can associate this poset structure on subobjects, and even better, sharpen the characerization by looking at kernels (at least in the case of groups and rings). However, my point was that the additional operation of multiplication, which is restricted to finite products (I guess I would say it has finite arity, but that's not exactly right), gives us an operation on the poset that looks like "finite unions" or "finite intersections".

More generally, for any sober space (and thus for $\mathrm{Spec}(R)$ for any ring $R$), the homeomorphism type is specified by the lattice of opens. The points of a sober space are in bijective correspondence with the completely prime filters on the lattice of open sets.
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Clark BarwickFeb 8 '10 at 12:42

@Harry : are by any chance you working your way towards a quantale?
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Tim PorterFeb 8 '10 at 17:25

@Tim, I didn't know that was a thing. I just looked it up, and it seems to be what I was talking about.
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Harry GindiFeb 8 '10 at 17:37

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There are even quantaloids and quantaloid enriched categories! Fun and relevant to discussions on the café perhaps.
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Tim PorterFeb 8 '10 at 18:41

Neat! But I mean, can't you just describe all quantale-looking things as monoidal categories?
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Harry GindiFeb 8 '10 at 21:00

I think Von Neumann might answer this question like this (as he once has): "One does not understand anything in mathematics one simply gets used to it." Then he may add, we have a natural functor from the category of topological spaces to the category of rings, it is an interesting natural phenomenon that this functor is strictly invertible if we restrict to suitable subcategories of the category of top spaces, and of the category of rings. This is Gelfand-Naimark.

One does not "understand" phenomena anymore then one understands why the universe exists.